-) when is the part of codimension 2 of the singular locus of a codimension one foliation on $P^n$ ($P^3$ actually) connected?

-) what conditions can we give on a codimension one foliation on $P^n$ to assure that its tangent sheaf it's split?

We (partially) solve this two problems by considering the Kupka scheme associated to a codimension one foliation on $P^n$ and proving that it is arithmetically Cohen-Macaulay. We did this by using the relation between the Kupka scheme and the unfoldings ideal associated to a codimension one foliation. We will review this relation and address the two questions above. Joint work with O. Calvo-Andrade and F. Quallbrunn.

Bibliography: O. Calvo-Andrade, A. Molinuevo, F. Quallbrunn, On the geometry of the singular locus of a codimension one foliation in $P^n$. Availabe at https://arxiv.org/abs/1611.03800